📐 High School Geometry & Trigonometry
At the high school level, we focus on the relationships between angles and sides, as well as the properties of circles and coordinate planes.
1. Right Triangle Trigonometry
When dealing with right triangles, we use the ratios of the sides relative to an angle $\theta$.
SOH CAH TOA:
- Sine: $\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}}$
- Cosine: $\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}}$
- Tangent: $\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}}$
2. Laws for Non-Right Triangles
For any triangle with sides $a, b, c$ and opposite angles $A, B, C$:
Law of Sines
The ratio of the length of a side to the sine of its opposite angle is constant. $$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$$
Law of Cosines
Useful for finding a missing side when you have two sides and the included angle (SAS). $$c^2 = a^2 + b^2 - 2ab \cos(C)$$
3. Coordinate Geometry
Formulas used for points $(x_1, y_1)$ and $(x_2, y_2)$ on a Cartesian plane.
- Distance Formula: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$
- Midpoint Formula: $M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$
- Slope ($m$): $m = \frac{y_2 - y_1}{x_2 - x_1}$
4. Circles & Conic Sections
Beyond area, we look at the equation of a circle and arc properties.
- Equation of a Circle: $(x - h)^2 + (y - k)^2 = r^2$ (where $(h, k)$ is the center).
- Arc Length ($s$): $s = r\theta$ (where $\theta$ is in radians).
- Sector Area ($A$): $A = \frac{1}{2}r^2\theta$
5. Advanced Volume (Pyramids, Cones, Spheres)
| Shape | Volume ($V$) | Surface Area ($SA$) |
|---|---|---|
| Cone | $V = \frac{1}{3}\pi r^2 h$ | $SA = \pi r^2 + \pi rl$ ($l$ = slant height) |
| Pyramid | $V = \frac{1}{3}Bh$ | $SA = B + \frac{1}{2}Pl$ ($P$ = perimeter of base) |
| Sphere | $V = \frac{4}{3}\pi r^3$ | $SA = 4\pi r^2$ |
💡 The Calculus Connection
As you move toward Calculus, remember that Volume is the integral of Area, and Area is the integral of Perimeter (or circumference).
Example: $\frac{d}{dr}(\text{Volume of Sphere}) = \frac{d}{dr}(\frac{4}{3}\pi r^3) = 4\pi r^2$ (Surface Area!)